Recent work has highlighted the role that abyssal mixing layers play in the circulation of the abyssal ocean [1]}, [2]}, [3]}, [4]}, [5]}, [6]}.
A starting point for understanding these dynamics of a stratified, rotating fluid overlying an inclined seafloor has been the canonical 1D theory first developed by [7]} and [8]}.
We have shown here, however, that the choice to set the cross-slope pressure gradient to zero in these dynamics eliminates important physics.
If instead a constraint is imposed on the vertically integrated cross-slope transport, which can be thought of as arising from the non-local context of the 1D column, and a barotropic cross-slope pressure gradient is allowed, rapid spin up and spin down of the interior along-slope flow can be captured.
With this transport constraint, a secondary cross-slope circulation can develop in the 1D framework, even if there are no lateral variations in the flow, and act on the interior flow.
These modified 1D dynamics accurately capture the mixing-generated spin up over an idealized 2D ridge, where the canonical 1D dynamics fail.
It can be hoped that these transport-constrained 1D dynamics can serve as a more reliable cornerstone for building a theory of the abyssal circulation than the canonical 1D system.
